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A298016 Coordination sequence of snub-632 tiling with respect to a hexavalent node. 22
1, 6, 12, 12, 24, 36, 24, 42, 60, 36, 60, 84, 48, 78, 108, 60, 96, 132, 72, 114, 156, 84, 132, 180, 96, 150, 204, 108, 168, 228, 120, 186, 252, 132, 204, 276, 144, 222, 300, 156, 240, 324, 168, 258, 348, 180, 276, 372, 192, 294, 396, 204, 312, 420, 216, 330, 444, 228, 348, 468, 240 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The snub-632 tiling in also called the fsz-d net. It is the dual of the 3.3.3.3.6 Archimedean tiling.

This is also called the "6-fold pentille" tiling in Conway, Burgiel, Goodman-Strauss, 2008, p. 288. - Felix Fröhlich, Jan 13 2018

REFERENCES

J. H. Conway, H. Burgiel and C. Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5.

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

C. Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, arXiv:1803.08530 [math.CO], March 2018.

N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

FORMULA

For n >= 1, let k=floor(n/3). Then a(3*k) = 12*k, a(3*k+1)=18*k+6, a(3*k+2)=24*k+12.

a(n) = 2*a(n-3) - a(n-6) for n >= 7.

G.f.: -(-x^6-12*x^5-12*x^4-10*x^3-12*x^2-6*x-1)/(x^6-2*x^3+1).

MAPLE

f:=proc(n) local k, r;

if n=0 then return(1); fi;

r:=(n mod 3); k:=(n-r)/3;

if r=0 then 12*k elif r=1 then 18*k+6 else 24*k+12; fi;

end;

[seq(f(n), n=0..80)];

MATHEMATICA

Join[{1}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {6, 12, 12, 24, 36, 24}, 60]] (* Jean-François Alcover, Apr 23 2018 *)

PROG

(PARI) Vec((1 + 6*x + 12*x^2 + 10*x^3 + 12*x^4 + 12*x^5 + x^6) / ((1 - x)^2*(1 + x + x^2)^2) + O(x^60)) \\ Colin Barker, Jan 13 2018

CROSSREFS

Cf. A298014, A298015.

List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

Sequence in context: A181798 A323332 A183688 * A055595 A132632 A223352

Adjacent sequences:  A298013 A298014 A298015 * A298017 A298018 A298019

KEYWORD

nonn,easy

AUTHOR

Chaim Goodman-Strauss and N. J. A. Sloane, Jan 11 2018

STATUS

approved

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Last modified January 17 18:37 EST 2019. Contains 319251 sequences. (Running on oeis4.)